Maximizing Rosh Hashanah Spiritually

Why does Rosh Hashanah, the day of judgment, come before Yom Kippur, when forgiveness should come first? And if it opens the Aseres Yemei Teshuva, why do we avoid even mentioning sin that day? The Ramban brings a Mechilta about a king approaching a city, and the residents who ask for his laws first. It frames a dispute among the Rishonim over the first of the Ten Commandments. Then comes the voice that tells us, "you'll only sin again," and what Hashem asks us to bring to the shofar instead.